Langevin theory
Within the room temperature region magnetic nanoparticles are typically superparamagnetic materials. A theoretical magnetization curve is calculated by the Langevin formula. The magnetic susceptibility of a material, commonly symbolized by χm, is equal to the ratio of the magnetization M within the material to the applied magnetic field strength H, or χm= M/H. A typical value given is initial susceptibility is calculated from the dΜ/dH (slope) of the initial magnetization curve:
χ = (∂M/∂H)i = μ0φMd2Vm/3kBT
Reference: The Langevin Formula for Describing the Magnetization Curve of a Magnetic Liquid S. V. D’yachenko and A. I. Zhernovoi, Solid State Volume 61, 835–1837, (2016)
http://doi.org/10.1134/S1063784216120112Description: μ0: permeability in free space (4π x 10-7), kB: Boltzmann constant, T: temperature, ρ: particle density, Md: bulk magnetization, MS: saturation magnetization of the bulk material, ms: saturation magnetization of the nanoparticles, φ: nanoparticle volume fraction (mS/MS), r: nanoparticle radius, Vm: nanoparticle volume
Chantrell Theory
For a ferrofluid with a lognormal distribution of particle size, a method to determine the median magnetic particle diameter (DM) and its standard deviation (σν) from the room temperature magnetisation curve, which differ from the 'physical size distribution' parameters obtained from electron microscope data.
DM = [18kBT/πMS √χin/3mSH0]1/3 & σV = 1/3[ln((3χin/mS)(1/H0))]1/2
Reference: Chantrell, R., Popplewell, J., & Charles, S. (1978). Measurements of particle size distribution parameters in ferrofluids. IEEE Transactions on Magnetics, 14(5), 975–977.
https://doi.org/10.1109/TMAG.1978.1059918Description: kB: Boltzmann constant, T: temperature, MS: saturation magnetization of the bulk material, ms: saturation magnetization of the nanoparticles, χin: initial susceptibility - Langevin theory, H0: magnetic field, DM: median magnetic particle diameter, σV: standard deviation from the room temperature magnetisation curve